Perpendicular-bias-field-dependent vortex-gyration eigenfrequency
Myoung-Woo Yoo, Ki-Suk Lee, Dong-Soo Han, and Sang-Koog Kim
Citation: Journal of Applied Physics 109, 063903 (2011); doi: 10.1063/1.3563561 View online: http://dx.doi.org/10.1063/1.3563561
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/109/6?ver=pdfcov Published by the AIP Publishing
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Perpendicular-bias-field-dependent vortex-gyration eigenfrequency
Myoung-Woo Yoo, Ki-Suk Lee, Dong-Soo Han, and Sang-Koog Kima)
Research Center for Spin Dynamics & Spin-Wave Devices, Nanospinics Laboratory, Research Institute of Advanced Materials, Department of Materials Science and Engineering, Seoul National University, Seoul 151-744, South Korea
(Received 13 January 2011; accepted 22 January 2011; published online 17 March 2011)
We found that the angular frequencyx0of vortex-core gyrations is controllable by the application of static perpendicular bias fieldsHpas studied by micromagnetic simulations and Thiele’s-approach-based quantitative interpretation. The observed linear dependence ofx0onHpcould be explained in terms of the dynamic variables of the vortex, the gyrovector constantG, and the potential stiffness constantj, for cases of negligible damping. Here we calculated the values ofGandjas a function ofHpdirectly from the simulation numerical data using Thiele’s equivalent force equations, providing a more correct understanding of the remarkable change of x0 with Hp. This micromagnetic-simulation-based quantitative analysis is a straightforward, accurate, and effective means of understanding vortex dynamics in nanoscale magnetic elements.VC 2011 American Institute of Physics.
[doi:10.1063/1.3563561]
I. INTRODUCTION
Magnetic vortex configurations have received a great deal of attention, not only because of the vortex structure’s unique fourfold ground states in magnetic nanodots of mi- crometer size or smaller1,2but also due to its nontrivial vor- tex-core reversal dynamics.3–7The fourfold vortex states are represented by combinations of either counter-clockwise (CCW) or clockwise (CW) in-plane curling magnetization and either up- or downward out-of-plane magnetization at the core.1Owing not only to the binary representation of the core orientation8 but also to the full degrees of freedom of the fourfold ground states,9such a stable vortex structure has been considered as a candidate for potential information stor- age applications.10 Furthermore, the existence of the vortex core, combined with the restoring force associated with nanodots’ finite lateral size, gives rise to in-plane rotating motions of the core around its equilibrium center position at the characteristic angular eigenfrequency x0 in either the CCW or CW angular rotation direction, so-named the gyro- tropic mode.11Such persistent vortex oscillations, with their high power output and narrow line widths, have been studied intensively recently for their promising application to micro- wave emission sources.12
It is well known that the rotation sense of the gyrotropic mode is determined by the core magnetization direction11 and that the value ofx0depends on the dot dimensions, that is, thickness-to-radius (L/R) ratio for the given material.13x0
typically falls within the 50–600 MHz range for L/R ratios between 0.01 and 0.12.13 The x0, then, is not controllable for a given dot. Finding a way to manipulate it with external parameters such as fields, meanwhile, is a challenge. Choi et al.14reported that the eigenfrequency can be modified by Oersted fields produced by an out-of-plane dc current, spe- cifically by varying the potential energy of the dots. Very
recently, Loubenset al.15reported an experimental observa- tion of the dependence ofx0on the strength of a static mag- netic field Hpapplied perpendicularly to the nanodot plane.
However, the underlying physics have yet to be quantita- tively unraveled. In this paper, we report, on the basis of micromagnetic simulations, that the value ofx0is controlla- bly variable linearly withHpin its sufficiently wide but lim- ited range. Furthermore, this work provides a more correct understanding of the linear dependence ofx0onHpas deter- mined by direct calculation of the Hp dependences of the dynamic variables of the vortex through Thiele’s equivalent force equation,16using the numerical simulation data.
II. MODELING AND MICROMAGNETIC SIMULATION In the present micromagnetic simulations, we used the OOMMF code17implementing the Landau–Lifshitz–Gilbert (LLG) equation18 of motion. We chose, for the model sys- tem, a permalloy (Py: Ni80Fe20) nanodisk of 300 nm radius and 30 nm thickness [Fig.1(a)]. The typical Py material pa- rameters (saturation magnetization Ms¼860 emu/cm3, exchange stiffness Aex¼1.3106erg/cm, with zero mag- netocrystalline anisotropy) were employed. The size of the individual cells was 2230 nm3, and the Gilbert damping constant was a¼0.01. Static magnetic fields were applied perpendicularly to the dot plane, and the strength varied fromHp¼ 3 to 3 kOe at intervals of 1 kOe.19The negative (positive) sign corresponds to the antiparallel (parallel) ori- entation of the core magnetization to the field direction.
III. RESULTS AND DISCUSSION
To examine howx0varies withHp, we used the follow- ing typical approach: first, we shifted the core from the cen- ter position by application of a static field ofHy¼100 Oe in theþy direction; then, during free relaxation after the in- plane field was turned off, the averaged oscillations of the mx¼Mx=Mscomponent (hmxi) were calculated as a function
a)Author to whom correspondence should be addressed. Electronic mail:
0021-8979/2011/109(6)/063903/4/$30.00 109, 063903-1 VC2011 American Institute of Physics
of time for differentHpvalues [Fig.1(b)]. Through fast Fou- rier transforms (FFTs) of the hmxi oscillations, we plotted FFT powers in the frequency domain as a function ofHp, as shown in Fig.1(c). The individual dominant frequency peaks represent x0=2p for a given Hp. The value of x0=2p at Hp¼0 is about 430 MHz, which increases monotonically to 560 MHz atHp¼ þ3 kOe and which decreases to 290 MHz atHp¼ 3 kOe. Thex0variation with Hp(closed circles), as can be seen in Fig.2, shows a linear dependence.
To understand such a linear response, we start with Thiele’s analytical approach.16 Assuming a steady-state motion with core velocityv, the governing equation of the gyrotropic mode is given as GvDv^ þ@WðXÞ=
@X¼0, where G¼ pG^z is the gyrovector with its con- stant G>0 and the core polarization p, and D^¼DI^is the damping tensor with the identity matrix I^and its constant D<0. The potential energy function is given as WðXÞ ¼Wð0Þ þ1=2jjXj2with the vortex-core position vec- tor X for a linear regime.11 The first term is the potential energy forX¼0, and the second, the potential energy of the intrinsic stiffness coefficientjfor displaced core positionX, as dominated by exchange and magnetostatic energies. Solv- ing this governing equation, the characteristic angular eigen- frequency is given as xD¼jG=ðG2þD2Þ with a certain damping D.20 For cases of negligible damping, that is, G jDj,xDbecomes approximatelyx0 ¼j=G.11Based on a two-vortex side surface charge free model,11,13,21Gandj can be expressed explicitly asG¼2pLMs=cwith the gyro-
magnetic ratio c, and j¼409pMs2L2=R when L=R1.
Accordingly,x0is explicitly given asx0¼209 MscL=R(Ref.
13) and hence is determined only by theL=Raspect ratio for a given material. As such, theHpdependence ofx0shown in Fig.2(closed circles) cannot be understood. Recently Lou- bens et al.15 reported an experimental result of the linear response, describing the result as an explicit expression of x0ðHpÞ ¼x0ð0Þ½1þpðHp=HsÞ, whereHsis the field to sat- urate the dot along the normal (see solid line in Fig.2). Our simulation results and the analytical expression were found to be in general agreement but with 14% difference. The analytical form was estimated from the relation ofx0¼j=G (Ref.11) for the caseHp¼0 by employing the estimated an- alytical forms GðHpÞ ¼Gð0Þ½1pðHp=HsÞ and jðHpÞ
¼jð0Þ½1 ðHp=HsÞ2.15However, such explicit expressions cannot properly be derived analytically due to the fact that because the application of Hp modifies the ground magnet- ization state of a vortex, the two-vortex side surface charge free model, which was used to derive analytical equations of GandjatHp¼0, cannot be applicable to the cases of appli- cation of perpendicular fields.
For a deeper or more correct understanding, the varia- tions ofGandjwithHpshould be calculated from the results of simulations in which variations of the vortex ground state withHpare accounted for. Our main idea is to use the equiva- lent force balance equation FGþFDþFW¼0, where each term corresponds to the gyrotropic force, the damping force, and the restoring force due to the potential energyW, respec- tively, as indicated by the symbols in the superscript. The val- ues of G and Dcan be extracted directly from those force terms FG¼pG^zv and FD¼D^v¼Dv, without any assumption, when the values of v, FG, and FD are known.
The force terms can be calculated from the simulation numer- ical data byFG¼P
fGi andFD¼P
fDi withfGi ¼ ðfi;xG;fi;yGÞ,
fDi ¼ ðfi;xD;fi;yDÞ,where each term for theith cell is given as
FIG. 1. (Color online) (a) Ground vortex state (left) and core-shifted state (right) with upward core magnetization and counter-clockwise (CCW) in-plane curling magnetization in Py disk of diameter 2R¼600 nm and thick- ness L¼30 nm. The colors and heights represent the local in- and out- of-plane magnetization components, respectively. (b) Averagemxcomponent hmxiover whole disk vs time under perpendicular bias fields of indicated strengthHp, during relaxation process after vortex core is shifted. (c) FFT power spectra for differentHpvalues, obtained from FFTs ofhmxioscillations overt¼0-100 ns range.
FIG. 2. (Color online)Hpdependence ofx0(closed circles), obtained from FFT power spectra [Fig.1(c)], compared with results forj(Hp)/G(Hp) (open circles) obtained from Hp-dependent G and j values (see Fig. 3), also obtained directly from simulation data. The solid line indicates the result of Eq.(2)reported in Ref.15.
063903-2 Yooet al. J. Appl. Phys.109, 063903 (2011)
fi;xG¼ ð1=cMs2ÞðMdM=dtÞ @M=@x;
fi;yG¼ ð1=cMs2ÞðMdM=dtÞ @M=@y; (1a) fi;xD¼ ða=cMsÞðdM=dtÞ @M=@x;
fi;yD¼ ða=cMsÞðdM=dtÞ @M=@y: (1b) The calculation results (open circles) forGas functions of Hp, using the approach explained in the preceding text, are shown in Fig. 3(a). The slope of DG=DHp, plotted in Fig.3(a), is about 9.81011ergs/(cm2kOe). The nega- tive sign indicates thatGdecreases with increasingHp. This is consistent with the fact thatGis determined by the relative magnitude of the out-of-plane magnetization of the core with respect to those around it.16This variation is in good agree- ment (within a 6% difference) with the explicit expression (solid line) GðHpÞ ¼Gð0Þ½1pðHp=HsÞ with Gð0Þ
¼2pLMs=c(Ref. 11) forc¼2p2.8 MHz/Oe,L¼30 nm, andHs¼10 kOe, as obtained for our nanodot dimensions.
The calculation results of D are shown in Fig. 3(b). D increases with increasingHpbut deviates slightly from the lin- ear response in the negativeHpregion.D¼ 2.14021011 ergs/cm2 at Hp¼0 is close to the value 2.30161011 ergs/cm2obtained from the analytic equation D¼ apMsL
½2þlnðR=RcÞ=c (Ref. 22) for R¼300 nm and L¼30 nm and assuming the vortex-core radiusRc¼15 nm. This equa- tion was derived for the case ofHp¼0 based on the model proposed by Usovet al.23TheG jDjresults plotted in Fig.
3indicate that the damping term only negligibly contributes to the eigenfrequency.
Next, theHpdependence of thejvalue was numerically calculated from the simulation data using WðXÞ ¼Wð0Þ þ1=2jjXj2. To calculate the total potential energyWversus
X¼|X|, the vortex core was shifted from the initial center position and then relaxed, as shown in Fig.1(b). The slope in the plot ofWversusX2accordingly gives rise toj=2, as seen in Fig.4(a). The resultantHp-dependentjvalues are shown in Fig. 4(b) (see open circles). The value of j atHp¼0 is estimated to2.635 erg/cm2, which value is smaller by 11%
than that of the analytical expression reported in Refs. 11and 13. This seems that the analytical expression has been derived assuming the two-vortices model11,13,21 that is dif- ferent from dynamic vortex configurations in the simula- tions. The j value decreases with increasing |Hp|, and the parabolic curve is somewhat asymmetric at about Hp¼0 in contrast to the explicit parabolic expression (solid line) of jðHpÞ ¼jð0Þ ½1 ðHp=HsÞ2 reported in Ref. 15. This explicit form was obtained for the assumption that the poten- tial energy profiles are the same for the different signs of þHpand –Hp. The asymmetry of our result originates from the difference in the mz profile of the vortex core between þHpand –Hp,shown in the inset of Fig.4(b). This appears to break the symmetry ofj at aroundHp¼0. The difference in x0between our simulation results (closed circles) and the ana- lytical estimation (solid line) in Ref.15can be ascribed mainly to the difference in j: j(3 kOe)¼2.232 erg/cm2, j(þ3 kOe)¼2.365 erg/cm2, from our simulation, and j(3 kOe)¼j(þ3 kOe)¼2.591 erg/cm2, from the analytical equa- tion. The core mz profiles not considered in the analytical expression would give rise to incorrect values ofjfor posi- tive and negativeHpvalues as can be evidenced by excellent agreements between the simulation results ofx0ðHpÞ(closed circles) and the results of jðHpÞ=GðHpÞ (open circles), shown in Fig.2.
FIG. 3. (Color online) Dependences of dynamic variablesGandDonHp. The open circles show the simulation results obtained throughG¼pjFGj=
jvj,D¼ jFDj=jvj. The solid line corresponds to the result obtained from an explicit expression ofGðHpÞ ¼Gð0Þ1pðHp=HsÞ
, as reported in Ref.15, whereHs¼10 kOe was used for our nanodot dimensions.
FIG. 4. (Color online) (a) Calculated total energyW(X) vsjXj2forHp¼0 kOe. The solid line indicates a linear fit to the simulation results (dots). (b) Dependence ofjonHp. The open circles indicate values obtained directly from the simulation results, while the solid line corresponds to the result for jðHpÞ ¼jð0Þ½1 ðHp=HsÞ2reported in Ref.15. The inset shows the com- parison of the core-regionmzprofiles forHp¼ þ3 and3 kOe.
IV. CONCLUSIONS
We studied, by micromagnetic simulations combined with an analytical approach, the dependence of the angular frequency of vortex core oscillations uponHp. The observed linear increase of the vortex eigenfrequency withHpcould be explained quantitatively in terms of theHpdependences of all of the vortex dynamic variables,G,j, andD, calcu- lated numerically from the micromagnetic simulation results.
This work shows how numerical simulations and their quan- titative interpretations promise a more correct understanding of vortex dynamics in nanodots.
ACKNOWLEDGMENTS
This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 20100000706).
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